I am trying to understand comorphisms of Lie pseudoalgebras from the point of view of fibred categories, but failing miserably so far. My question would be:

*Is there a (op)fibration $\mathrm{LiePs} \to \mathrm{Alg}$ of the category of Lie pseudoalgebras in the sense of (op)fibred categories such that comorphisms of Lie pseudoalgebras are morphisms in the dual fibration $\mathrm{LiePs}^*$?*

Let me be a bit more precise: Lie pseudoalgebras (aka Lie-Rinehart algebras or others) can be seen as the algebraic counterparts of Lie algebroids.

Definition: (Lie Pseudoalgebra)

ALie pseudoalgebraconsists of a commutative algebra $\mathcal{A}$ and a Lie algebra $\mathfrak{g}$, such that $\mathfrak{g}$ is an $\mathcal{A}$-module and $\mathcal{A}$ acts on $\mathfrak{g}$ by derivations, i.e. we have a Lie algebra morphism $\rho \colon \mathfrak{g} \to \mathrm{Der}(\mathcal{A})$ which is also an $\mathcal{A}$-module morphism and we have $$ [\xi_1, a \xi_2] = a[\xi_1,\xi_2] + \rho(\xi_1)(a)\xi_2$$ for $\xi_1, \xi_2 \in \mathfrak{g}$, $a \in \mathcal{A}$.

For Lie pseudoalgebras there is a quite obvious notion of morphism.

Definition: (Morphism of Lie pseudoalgebras)

A morphism of Lie pseudoalgebras $(\mathcal{A},\mathfrak{g})$ and $(\mathcal{B},\mathfrak{h})$ consists of an algebra morphism $\phi \colon \mathcal{A} \to \mathcal{B}$ and a module morphism $\Phi \colon \mathfrak{g} \to \mathfrak{h}$ along $\phi$ which is also a Lie algebra morphism and $$ \phi(\rho_{\mathcal{A}}(\xi)a) = \rho_{\mathcal{B}}(\Phi(\xi))\phi(a)$$ holds for $\xi \in \mathfrak{g}$, $a \in \mathcal{A}$.

Let us denote the category of Lie pseudoalgebras by $\mathrm{LiePs}$. Then there is an obvious functor $\mathrm{LiePs} \to \mathrm{Alg}$ by mapping a Lie pseudoalgebra $(\mathcal{A},\mathfrak{g})$ to the algebra $\mathcal{A}$ and a morphism $(\Phi,\phi)$ to the corresponding algebra morphism $\phi$.

*Question 1:*
Is $\mathrm{LiePs} \to \mathrm{Alg}$ a (op)fibration of categories?

If so: for (op)fibred categories there is a notion of a dual (op)fibration, see e.g. A.Kock; The dual fibration in elementary terms, whose morphisms can be understood as comorphisms of the original objects. And there is a notion of comorphism of Lie pseudoalgebras, see e.g. Z. Chen,Z.-J. Liu; On (co-)morphisms of Lie pseudoalgebras and groupoids.

*Question 2:* Do comorphisms of Lie pseudoalgebras agree with morphisms in the dual fibration
$\mathrm{LiePs}^* \to \mathrm{Alg}$?